<h2>题目编号 : 247</h2>
<div style="color:#666;font-size:80%;">29 May 2009</div><br />
<div class="problem_content">
<p>Consider the region constrained by 1 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>x</var> and 0 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>y</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <img src="" style="display:none;" alt="^(" /><sup>1</sup><img src="" style="display:none;" alt=")" />/<img src="" style="display:none;" alt="_(" /><sub><var>x</var></sub><img src="" style="display:none;" alt=")" />.
</p><p>
Let S<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> be the largest square that can fit under the curve.<br />
Let S<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> be the largest square that fits in the remaining area, and so on. <br />
Let the <i>index</i> of S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> be the pair (left, below) indicating the number of squares to the left of S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> and the number of squares below S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" />.
</p>
<div style="text-align:center;">
<img src="project/images/p_247_hypersquares.gif" alt="" /></div>
<p>
The diagram shows some such squares labelled by number. <br />
S<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> has one square to its left and none below, so the index of S<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> is (1,0).<br />
It can be seen that the index of S<img src="" style="display:none;" alt="_(" /><sub>32</sub><img src="" style="display:none;" alt=")" /> is (1,1) as is the index of S<img src="" style="display:none;" alt="_(" /><sub>50</sub><img src="" style="display:none;" alt=")" />. <br />
50 is the largest <var>n</var> for which the index of S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> is (1,1).
</p>
<p>
What is the largest <var>n</var> for which the index of S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> is (3,3)?
</p>


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